We devise a new undecimated wavelet thresholding for denoising images corrupted by additive Gaussian white noise. The first key point of our approach is the use of a linearly parameterized pointwise thresholding function. The second key point consists in optimizing the parameters globally by minimizing Stein's unbiased MSE estimate (SURE) directly in the image-domain, and not separately in the wavelet subbands. Amazingly, our method gives similar results to the best state-of-the-art algorithms, despite using only a simple pointwise thresholding function; we demonstrate it in simulations over a wide range of noise levels for a representative set of standard grayscale images.